[tex] \displaystyle\lim_{x \to 1} (1-x)\cdot \frac{sin \frac{\pi x}{2} }{cos \frac{\pi x}{2}} =\\
=\lim_{x \to 1} sin \frac{\pi x}{2} \cdot \lim_{x \to 1} \frac{1-x }{cos \frac{\pi x}{2}}=\\
=\lim_{x \to 1} sin \frac{\pi x}{2} \cdot \lim_{x \to 1} \frac{1-x }{cos \frac{\pi x}{2}}=\\
=1 \cdot \lim_{x \to 1} \frac{1-x }{cos \frac{\pi x}{2}}=\\
= \lim_{x \to 1} \frac{1-x }{cos \frac{\pi x}{2}}=-Notam\ x-1=t\\
= \lim_{t \to 0} \frac{-t}{cos( \frac{\pi t}{2}+ \frac{\pi}{2} )} =\\
[/tex]
[tex]\displaystyle= \lim_{t \to 0} \frac{-t}{cos( \frac{\pi t}{2}+ \frac{\pi}{2} )} =\\
= \lim_{t \to 0} \frac{-t}{-sin \frac{\pi t}{2} } =\\
=\lim_{t \to 0} \frac{t}{sin \frac{\pi t}{2} } =\\
= \frac{1}{ \frac{\pi}{2} } =\\
=\frac{2}{\pi} [/tex]
